Finite Math Examples

$10 x - 7 y = - 46$ , $- 4 x - 5 y = 34$

Step 1

Solve for $x$ in $10 x - 7 y = - 46$ .

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Step 1.1

Add $7 y$ to both sides of the equation.

$10 x = - 46 + 7 y$

$- 4 x - 5 y = 34$

Step 1.2

Divide each term in $10 x = - 46 + 7 y$ by $10$ and simplify.

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Step 1.2.1

Divide each term in $10 x = - 46 + 7 y$ by $10$ .

$\frac{10 x}{10} = \frac{- 46}{10} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $10$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{10 x}{10} = \frac{- 46}{10} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 46}{10} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = \frac{- 46}{10} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = \frac{- 46}{10} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 46$ and $10$ .

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Step 1.2.3.1.1.1

Factor $2$ out of $- 46$ .

$x = \frac{2 (- 23)}{10} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $2$ out of $10$ .

$x = \frac{2 \cdot - 23}{2 \cdot 5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot - 23}{2 \cdot 5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$x = \frac{- 23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = \frac{- 23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = \frac{- 23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 1.2.3.1.2

Move the negative in front of the fraction.

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 4 x - 5 y = 34$

Step 2

Replace all occurrences of $x$ with $- \frac{23}{5} + \frac{7 y}{10}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $- 4 x - 5 y = 34$ with $- \frac{23}{5} + \frac{7 y}{10}$ .

$- 4 (- \frac{23}{5} + \frac{7 y}{10}) - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $- 4 (- \frac{23}{5} + \frac{7 y}{10}) - 5 y$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Apply the distributive property.

$- 4 (- \frac{23}{5}) - 4 \frac{7 y}{10} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.2

Multiply $- 4 (- \frac{23}{5})$ .

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Step 2.2.1.1.2.1

Multiply $- 1$ by $- 4$ .

$4 (\frac{23}{5}) - 4 \frac{7 y}{10} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.2.2

Combine $4$ and $\frac{23}{5}$ .

$\frac{4 \cdot 23}{5} - 4 \frac{7 y}{10} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.2.3

Multiply $4$ by $23$ .

$\frac{92}{5} - 4 \frac{7 y}{10} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$\frac{92}{5} - 4 \frac{7 y}{10} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.3

Cancel the common factor of $2$ .

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Step 2.2.1.1.3.1

Factor $2$ out of $- 4$ .

$\frac{92}{5} + 2 (- 2) (\frac{7 y}{10}) - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.3.2

Factor $2$ out of $10$ .

$\frac{92}{5} + 2 \cdot (- 2 \frac{7 y}{2 \cdot 5}) - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.3.3

Cancel the common factor.

$\frac{92}{5} + 2 \cdot (- 2 \frac{7 y}{2 \cdot 5}) - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.3.4

Rewrite the expression.

$\frac{92}{5} - 2 \frac{7 y}{5} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$\frac{92}{5} - 2 \frac{7 y}{5} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.4

Combine $- 2$ and $\frac{7 y}{5}$ .

$\frac{92}{5} + \frac{- 2 (7 y)}{5} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.5

Multiply $7$ by $- 2$ .

$\frac{92}{5} + \frac{- 14 y}{5} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.1.6

Move the negative in front of the fraction.

$\frac{92}{5} - \frac{14 y}{5} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$\frac{92}{5} - \frac{14 y}{5} - 5 y = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.2

To write $- 5 y$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{92}{5} - \frac{14 y}{5} - 5 y \cdot \frac{5}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.3

Combine $- 5 y$ and $\frac{5}{5}$ .

$\frac{92}{5} - \frac{14 y}{5} + \frac{- 5 y \cdot 5}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.4

Combine the numerators over the common denominator.

$\frac{92}{5} + \frac{- 14 y - 5 y \cdot 5}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.5

Combine the numerators over the common denominator.

$\frac{92 - 14 y - 5 y \cdot 5}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.6

Multiply $5$ by $- 5$ .

$\frac{92 - 14 y - 25 y}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 2.2.1.7

Subtract $25 y$ from $- 14 y$ .

$\frac{92 - 39 y}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$\frac{92 - 39 y}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$\frac{92 - 39 y}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$\frac{92 - 39 y}{5} = 34$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3

Solve for $y$ in $\frac{92 - 39 y}{5} = 34$ .

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Step 3.1

Multiply both sides by $5$ .

$\frac{92 - 39 y}{5} \cdot 5 = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{92 - 39 y}{5} \cdot 5$ .

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Step 3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{92 - 39 y}{5} \cdot 5 = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.2.1.1.1.2

Rewrite the expression.

$92 - 39 y = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$92 - 39 y = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.2.1.1.2

Reorder $92$ and $- 39 y$ .

$- 39 y + 92 = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 39 y + 92 = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 39 y + 92 = 34 \cdot 5$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $34$ by $5$ .

$- 39 y + 92 = 170$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 39 y + 92 = 170$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 39 y + 92 = 170$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Move all terms not containing $y$ to the right side of the equation.

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Step 3.3.1.1

Subtract $92$ from both sides of the equation.

$- 39 y = 170 - 92$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.3.1.2

Subtract $92$ from $170$ .

$- 39 y = 78$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$- 39 y = 78$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.3.2

Divide each term in $- 39 y = 78$ by $- 39$ and simplify.

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Step 3.3.2.1

Divide each term in $- 39 y = 78$ by $- 39$ .

$\frac{- 39 y}{- 39} = \frac{78}{- 39}$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $- 39$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{- 39 y}{- 39} = \frac{78}{- 39}$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.3.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{78}{- 39}$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$y = \frac{78}{- 39}$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$y = \frac{78}{- 39}$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Divide $78$ by $- 39$ .

$y = - 2$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$y = - 2$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$y = - 2$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$y = - 2$

$x = - \frac{23}{5} + \frac{7 y}{10}$

$y = - 2$

$x = - \frac{23}{5} + \frac{7 y}{10}$

Step 4

Replace all occurrences of $y$ with $- 2$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - \frac{23}{5} + \frac{7 y}{10}$ with $- 2$ .

$x = - \frac{23}{5} + \frac{7 (- 2)}{10}$

$y = - 2$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- \frac{23}{5} + \frac{7 (- 2)}{10}$ .

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Step 4.2.1.1

Multiply $7$ by $- 2$ .

$x = - \frac{23}{5} + \frac{- 14}{10}$

$y = - 2$

Step 4.2.1.2

Simplify each term.

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Step 4.2.1.2.1

Cancel the common factor of $- 14$ and $10$ .

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Step 4.2.1.2.1.1

Factor $2$ out of $- 14$ .

$x = - \frac{23}{5} + \frac{2 (- 7)}{10}$

$y = - 2$

Step 4.2.1.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1.2.1

Factor $2$ out of $10$ .

$x = - \frac{23}{5} + \frac{2 \cdot - 7}{2 \cdot 5}$

$y = - 2$

Step 4.2.1.2.1.2.2

Cancel the common factor.

$x = - \frac{23}{5} + \frac{2 \cdot - 7}{2 \cdot 5}$

$y = - 2$

Step 4.2.1.2.1.2.3

Rewrite the expression.

$x = - \frac{23}{5} + \frac{- 7}{5}$

$y = - 2$

$x = - \frac{23}{5} + \frac{- 7}{5}$

$y = - 2$

$x = - \frac{23}{5} + \frac{- 7}{5}$

$y = - 2$

Step 4.2.1.2.2

Move the negative in front of the fraction.

$x = - \frac{23}{5} - \frac{7}{5}$

$y = - 2$

$x = - \frac{23}{5} - \frac{7}{5}$

$y = - 2$

Step 4.2.1.3

Combine fractions.

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Step 4.2.1.3.1

Combine the numerators over the common denominator.

$x = \frac{- 23 - 7}{5}$

$y = - 2$

Step 4.2.1.3.2

Simplify the expression.

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Step 4.2.1.3.2.1

Subtract $7$ from $- 23$ .

$x = \frac{- 30}{5}$

$y = - 2$

Step 4.2.1.3.2.2

Divide $- 30$ by $5$ .

$x = - 6$

$y = - 2$

$x = - 6$

$y = - 2$

$x = - 6$

$y = - 2$

$x = - 6$

$y = - 2$

$x = - 6$

$y = - 2$

$x = - 6$

$y = - 2$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- 6, - 2)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(- 6, - 2)$

Equation Form:

$x = - 6, y = - 2$

Step 7